Objective evaluation of linearization procedures in nonlinear homogenization: a methodology and some implications on the accuracy of micromechanical schemes

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Objective evaluation of linearization procedures in nonlinear homogenization: a methodology and some implications on the accuracy of micromechanical schemes

Amna Rekik, François Auslender, Michel Bornert, André Zaoui

To cite this version:

Amna Rekik, François Auslender, Michel Bornert, André Zaoui. Objective evaluation of linearization

procedures in nonlinear homogenization: a methodology and some implications on the accuracy of

micromechanical schemes. International Journal of Solids and Structures, Elsevier, 2007, 44 (10),

pp.3468-3496. �10.1016/j.ijsolstr.2006.10.001�. �hal-00111454�

Objective evaluation of linearization procedures in nonlinear homogenization: A methodology and some implications

on the accuracy of micromechanical schemes

Amna Rekik ^a , Franc¸ois Auslender ^a,b,* , Michel Bornert ^a , Andre´ Zaoui ^a

a

Laboratoire de Me´canique des Solides, E´cole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

b

Laboratoire de Me´canique et d’Inge´nieries, Universite´ Blaise Pascal/IFMA, BP 265, 63175 Aubie`re, France

A systematic methodology for an accurate evaluation of various existing linearization procedures sustaining mean fields theories for nonlinear composites is proposed and applied to recent homogenization methods. It relies on the analysis of a periodic composite for which an exact resolution of both the original nonlinear homogenization problem and the linear homogenization problems associated with the chosen linear comparison composite (LCC) with an identical microstructure is possible. The effects of the sole linearization scheme can then be evaluated without ambiguity. This methodology is applied to three different two-phase materials in which the constitutive behavior of at least one constituent is nonlinear elastic (or viscoplastic): a reinforced composite, a material in which both phases are nonlinear and a porous material. Com- parisons performed on these three materials between the considered homogenization schemes and the reference solution bear out the relevance and the performances of the modified second-order procedure introduced by Ponte Castan˜eda in terms of prediction of the effective responses. However, under the assumption that the field statistics (first and second moments) are given by the local fields in the LCC, all the recent nonlinear homogenization procedures still fail to provide an accurate enough estimate of the strain statistics, especially for composites with high contrast.

Keywords: Continuum mechanics; Homogenization; Nonlinear behavior; Linearization; Secant; Affine; Second-order

1. Introduction

Most of the nonlinear homogenization procedures for heterogeneous materials implicitly or explicitly rely on two separates stages. The first one consists in approximating the actual nonlinear behavior of the individual constituents of the composite by linear constitutive equations, through a specific linearization procedure, so as

*

Corresponding author. Tel.: +33 169333319; fax: +33 169333026.

E-mail addresses: rekik@lms.polytechnique.fr (A. Rekik), auslende@lms.polytechnique.fr (F. Auslender), bornert@lms.polytechnique.

fr (M. Bornert), zaoui@lms.polytechnique.fr (A. Zaoui).

to define a fictitious elastic or thermoelastic composite usually referred to as the linear comparison composite

‘‘LCC’’ (Ponte Castan˜eda, 1991). In the second stage, the overall as well as the local responses of this fictitious LCC are approximated by a linear homogenization model appropriate for the microstructure of the LCC. If in each stage, the derivation ensures that the result is an upper bound for the effective properties, the obtained global estimate is a bound as well (Ponte Castan˜eda, 1991) and its interpretation is clear. However in such a case, the predictions of the model might be far away from the exact response of the composite and thus be of limited practical interest. A less rigorous but more efficient linearization scheme might then be preferred and has to be selected among the numerous classical or more recently proposed formulations, such as Hill’s incre- mental scheme (Hill, 1965), the classical (Hutchinson, 1976; Berveiller and Zaoui, 1979) or modified (Ponte Castan˜eda, 1991; Suquet, 1995) secant approaches, the affine formulation (Masson and Zaoui, 1999; Masson et al., 2000) and its variants (Ponte Castan˜eda, 2002; Chaboche and Kanoute´, 2003), the second-order proce- dures (Ponte Castan˜eda, 1996, 2002) or the Lahellec and Suquet procedure (Lahellec and Suquet, 2004). These models are based on more or less sophisticated derivations and generate various predictions, which deserve an objective and systematic comparison to each other and with exact solutions, both at the local and global levels to appreciate their respective merits and limitations. It is the purpose of the present study to provide a meth- odology for such a systematic and objective comparison, from which, in addition, guidelines for improved for- mulations might be deduced.

A first and usual evaluation of the various linearization procedures consists in comparing their overall predictions with the few available (mostly upper) bounds, as in some previous studies (Gilormini, 1996).

However, such bounds are often not sufficiently sharp for a precise evaluation, except in some specific sit-

uations (Bornert and Ponte Castan˜eda, 1998). For weakly inhomogeneous composites, the predictions may

be compared with the expansions of the effective potential (Ponte Castan˜eda and Suquet, 1995, 1998) with

respect to the contrast, at various orders. In more general situations, accurate evaluations of the various

nonlinear homogenization procedures can be obtained by means of comparisons with full-field numerical

solutions of the initial nonlinear problem. To this end, two different approaches may be considered. For

the first one, numerical simulations are carried out on large windows of simulated microstructures (Mou-

linec and Suquet, 2003) supposed to depict the random microstructures addressed by the linear mean-field

theories used for the homogenization of the LCC. Such an approach induces computational difficulties due

to the large size of the numerical systems to work out, as well as issues relative to the representativeness

of the generated microstructures, the appropriate averaging of the results and the choice of particular

boundary conditions, both in the linear (Kanit et al., 2003) case and, even more critically, in the nonlinear

one. This approach implies a large computational expense and is in practice often restricted to two-dimen-

sional problems. Moreover, when three dimensional simulations are carried out, the current computer lim-

itations require to restrict the number of comparisons and thus forbid to thoroughly explore all

combinations of the parameters of the particular problem under consideration (Bilger et al., 2005). The

second approach consists in performing comparisons between the nonlinear homogenization procedures

and exact solutions on simple periodic microstructures. The main advantage of this approach lies in a rea-

sonable computational expense as well as an easier control of the numerical accuracy of the results (Chab-

oche and Kanoute´, 2003; Suquet, 1995). However, such an approach is questionable since the comparison

between the reference numerical solution and predictions from a nonlinear mean field theory based on a

linear homogenization model for a random LLC can be strongly altered by the fact that the compared

procedures do not address the same microstructure. They are also distorted by the use of linear closed-

form estimates—which are not exact results—to evaluate the effective properties of the LCC derived by

the linearization procedure of the mean field models. Such linear estimates may not be sufficiently accurate

and can provide crude approximations for complex microstructures or when the mechanical contrast

between constituents is high. Therefore, comparisons carried out in such a framework may lead to poten-

tially ambiguous results because of the addition of two approximations, stemming from the linearization

procedure and the linear homogenization scheme, which may be cumulative or compensative. Finally,

comparisons of predictions of nonlinear homogenization schemes may of course also be compared to

experimental data. However, such comparisons are very difficult since they combine approximations at

various levels: the actual nonlinear constitutive law of the constituents, the unavoidable simplification

of the microstructure used in the models with respect to the real one, the experimental scatters on the

evaluation of microstructural parameters such as volume fractions and of the overall responses as well as the complexity of local stress or strain measurements (Bornert et al., 1994). For theses reasons, at the present stage, ‘‘numerical experiments’’ are strongly preferred for such comparisons, since their inputs are known exactly and their outputs can be controlled. The main objective of this work is to remove their aforementioned limitations in order to be left with an objective evaluation of the sole linearization methods.

To this end, a new methodology which relies on an exact treatment of both the nonlinear and the linear homogenization problems is developed. Various nonlinear homogenization formulations are compared, main- ly with regard to their predictions in terms of overall responses. In addition, a preliminary evaluation of the local predictions is performed, restricted to the first and second moments of the local strain field, for the sake of brevity. At this stage, it is worth noting that, as in earlier works including very recent ones (e.g., Idiart et al., 2006), it is assumed that the local fields predicted by the nonlinear homogenization schemes are given by those in the LCC. This is consistent with the assumptions of classical formulations based on local fields averaging but can lead to some inconsistencies when overall responses are derived from evaluations of effective potentials (Masson et al., 2000; Lahellec and Suquet, 2004). As shown in a very recent paper by Idiart and Ponte Cas- tan˜eda (2006), not yet available at the time of submission of the present contribution, this limitation can be removed according to the general theory developed by these authors, which enables to extract in a consistent way the statistics of the local fields in the nonlinear composite from the knowledge of the effective potential of suitably perturbated composites. The corresponding correction terms, giving the discrepancy between the sta- tistics on the LCC and those on the nonlinear composite, are not taken into account in the present study.

Accordingly, the conclusions regarding the efficiency in terms of local predictions of these models should be considered as restricted to their simplified classical ‘‘LCC-based local interpretation’’. This will be discussed again with more details later on.

The structure of this paper is as follows. The methodology to evaluate the various linearization treat- ments sustaining nonlinear homogenization procedures is presented in Section 2. Its implementation is depicted in Section 3 where a complete description of the linear numerical elastic or thermoelastic homog- enization scheme and its connections with the linearization formulations is given. The main assumptions of the various linearization schemes that will be compared are recalled at the end of this section. Namely, these models are the classical secant scheme (referred to as SEC), the variational procedure (VAR), the original affine formulation (AFF-ANI), its isotropic simplification (AFF-ISOT), the original (SOE-1) and improved (SOE-2) second-order procedures, and finally the Lahellec and Suquet (LS) formulation.

The even more recent formulations such as those proposed by Idiart and Ponte Castan˜eda (2005) and Idi- art et al. (2006) have not been considered in the present work. On the other hand, Hill’s original incre- mental procedure is not considered either since its limitations have already been established earlier (Gilormini, 1996); its isotropic simplification, recently reassessed (Doghri and Ouaar, 2003), is not evalu- ated either since it is very close to the classical secant formulation, both schemes coinciding exactly when the local constitutive laws are power-laws with same exponent (Hutchinson, 1976). Similarly, the popular viscoplastic self-consistent model (Molinari et al., 1987; Lebensohn and Tome´, 1993), based on a tangent approximation of the nonlinear local constitutive law, has been omitted since it does not clearly separate the homogenization of a LCC from the linearization procedure and is restricted to random polycrystals.

The presented methodology is carried out in Section 4 on three different two-phase materials: a nonlinear matrix reinforced by linear elastic particles (referred to in the sequel as the ‘‘reinforced composite’’), a composite in which both phases exhibit a nonlinear behavior (referred to as the ‘‘two-phase material’’) and a porous material. Based on the numerous comparisons performed both at the global and local scales, a comparative evaluation of the various linearization procedures is finally proposed and the conclusions are summarized in Section 5.

The tensor notation used herein is a fairly standard notation. The orders of the tensors are clear when taken

in context. Products containing dots denote summation over repeated latin indices. For example, L : e =

L

ijkl

e

kl

e

i

e

j

and E :: F = E

ijkl

F

klij

where (e

i

, i = 1, 3) is a time-independent orthogonal cartesian basis and

denotes the tensorial product. Cylindrical coordinates will be used as well, with e

r

, e

h

, e

z

= e

3

being the unit

vectors of the orthogonal cylindrical basis and u

r

, u

h

, u

z

the cylindrical coordinate components of a 3D vector

u.

2. Proposed methodology 2.1. Principle

The methodology and the numerical tool which are proposed in this paper enable a systematic and accurate evaluation of the linearization procedures without suffering from the limitations mentioned in introduction.

We address a problem where the exact nonlinear solution, regarded as the reference solution, may be comput- ed with high accuracy at a low computational expense. Moreover, the homogenization of the LCC, which retains the microstructure of the nonlinear composite, is accurately computed by the same numerical tool as the one handling the nonlinear composite. Thereby, the difficulties related to the numerical approximations, to the change of microstructure as well as to the approximations induced by the linear estimates of the LCC are avoided. The effect of the sole linearization procedure may be assessed without any ambiguity.

The chosen microstructure is periodic and classical finite element techniques associated with periodic homogenization are used to solve both the initial nonlinear problem and the linear problem associated with the LCC. This methodology can be adopted for any type of local nonlinear constitutive relations, any number of phases and any microstructure, since large unit cells, able to mimic a random microstructure, may in prin- ciple be used. However, in order to restrict the computational expense, only small unit cells which describe sim- ple periodic composites will be used as a first application. Because of this practical restriction, the proposed methodology cannot evaluate exhaustively all situations. It however allows to handle large classes of problems, since various choices of local constitutive laws, contrasts between constituents, volume fractions, local geom- etries or loadings are possible. It thereby extends considerably the earlier comparisons of this type and, most importantly, provides unambiguous comparisons. In addition, since the linearization procedures are often sup- posed to apply to any type of microstructure, they should also work for periodic ones and can therefore be test- ed in this case. Let us also stress the fact that even if the effective properties and the heterogeneity of the local fields are quantitatively different in a random microstructure and a periodic one with similar constituents, they are qualitatively close so that comparisons performed between models for periodic microstructures should be sufficiently representative of what might be observed on random ones. Note finally that a similar approach has independently been used by Lahellec and Suquet (2004) to evaluate the specific model proposed by these authors, as well as by Moulinec and Suquet (2004) for a comparison between the classical and the modified secant linearization procedures. In both these contributions, only a limited number of models have been com- pared and comparisons were restricted to incompressible two-dimensional microstructures made of fiber rein- forced matrices loaded in their transverse plane, the response of which is characterized by a one-dimensional relation. In the present work, three dimensional microstructures with a wider range of phase contrasts are con- sidered, including porous materials exhibiting a two-dimensional response. An additional fundamental differ- ence between the proposed methodologies can be noticed: in these earlier works, the numerical simulations used for the evaluations of both the exact nonlinear behavior and the linear properties of the LCC aimed at repro- ducing complex random microstructures made of randomly distributed fibers. Expensive computational tools had to be used and statistical averages over results obtained with various realizations of complex unit cell had to be taken, giving rise to aforementioned corresponding questions relative to numerical and statistical conver- gence of the predictions. In the present proposed methodology, the explicit choice of simple periodic micro- structures intrinsically avoids such difficulties and allows one to explore more exhaustively the performances of many formulations, under various conditions, at a very low numerical cost. This is at the price of the anisot- ropy of the properties of both the nonlinear medium and the LCC, which can however easily be dealt with.

2.2. The nonlinear periodic composite

As a first illustration of the proposed methodology, we consider composites with isotropic nonlinear elastic

constituents, the behavior of which is governed by a single potential or strain energy function w(e), according

to r ¼ f ðeÞ ¼

^ow_oe

ðeÞ where r and e are the local stress and strain fields, respectively, in the framework of small

strains. Note that this case is similar to the rate problem for a nonlinear viscoplastic composite. For the sake

of simplicity, the studied material reduces to a periodic two-phase composite, made of aligned spherical inclu-

sions (particles or pores) embedded in a matrix. The inclusions are distributed on a hexagonal network in the

transverse plane and are aligned along the third direction, so that a cylinder with a hexagonal basis and a sin- gle spherical inclusion can be used as unit cell. All the loadings considered in this study are axially symmetric along the third direction. In a first and classical approximation (Michel et al., 2001), this hexagonal cell can be replaced by a cylindrical cell of height 2H with a circular basis of diameter 2R. Because of the invariance with respect to any rotation along the third direction of the geometry of the cell, of the constitutive relations of the phases and of the applied load, the 3D initial problem reduces to a 2D axially symmetric problem which can be solved at a very low numerical cost.

For numerical applications, the constitutive law r = f

r

(e) followed by phase r is a Ramberg–Osgood type relation with or without a threshold, an initial isotropic elastic behavior and a nonzero compressible part, which reads

e ¼ e

m

i þ K : e r ¼ r

m

i þ K : r

K : e ¼ 3e

eq

2r

eq

K : r ð1Þ

with

e

m

¼ r

m

3k

^r

and e

eq

¼ g

_r

ðr

eq

Þ ¼ r

eq

3l

^r_e

þ e

0

Pðr

eq

r

^r_y

Þ r

^r₀

ⁿ

: ð2Þ

In these expressions, k

^r

; l

^r_e

; r

^r₀

and r

^r_y

are the elastic bulk modulus, the elastic shear modulus, the flow stress and the threshold stress of the phase r, respectively. The parameter m = 1/n is the work-hardening parameter (n is the nonlinearity exponent) satisfying 0 6 m 6 1,

₀

is a reference strain and P(a) is the positive part of a.

The von Mises equivalents of the strain and the stress tensors are defined as usually by e

eq

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 3

e : K : e

q and

r

eq

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3

2

r : K : r q

, while r

m

¼

¹₃

i : r and e

m

¼

¹₃

i : e are their respective spherical parts. The tensors i and I are second and fourth-order identity tensors, and J ¼

¹₃

i i and K = I J are the usual projectors on the sub- spaces of purely spherical or deviatoric second-order tensors. The nonlinearity exponent n is the same in both phases, even if more general situations could be considered. Unlike usually assumed, l

^r_e

and k

^r

are finite in order to avoid practical numerical difficulties related to infinite tangent and secant moduli at e = 0 and to material incompressibility. We assume a perfect bonding between the inclusion and the matrix so that the dis- placement and surface traction are continuous at the interface.

2.3. Overall and local response under axial load

In the ensuing calculations, the unit cell is submitted to an axisymmetric macroscopic deformation

e ¼ e

m

i þ e

eq

^ e; ð3Þ

where ^ e ¼ ðK : eÞ=e

eq

¼ e

₃

e

₃

¹₂

ðe

1

e

₁

þ e

₂

e

₂

Þ. In the considered small strain formalism, e is the average of the local strain, say e ¼ hei, where the notation hCi ¼

_jV¹_j

R

V

CðxÞdx denotes the mean value of C over the cylin- drical unit cell V, whose measure is jVj = 2pR

²

H. Likewise and for a later use, hCi

^r

¼

_jV¹rj

R

V^r

CðxÞdx denotes the average of C over phase r, which occupies the domain V

^r

in the unit cell. Note that in the sequel e

m

and e

eq

are always positive quantities. The ratio s

e

¼ e

m

=e

eq

characterizes the macroscopic strain triaxiality ratio.

The local stress r and strain e fields in the unit cell are solution of the following set of equations (Michel et al., 2001)

uðxÞ ¼ e x þ u

ðxÞ 8x 2 V and u

# on oV

eðuðxÞÞ ¼

¹₂

ðruðxÞ þ

^t

ruðxÞÞ ¼ e þ eðu

ðxÞÞ; 8x 2 V divðrÞ ¼ 0; 8x 2 V and r n # on oV

rðxÞ ¼ P

r¼m;p

v

^r

ðxÞf

r

ðeðxÞÞ; 8x 2 V 8 >

> >

> >

<

> >

> >

> :

ð4Þ

where v

^r

(x) is the characteristic function of phase r and where the general notations # and # mean that the

fluctuating part of the displacement vector u

^*

and the surface traction r Æ n (n being the outer unit normal) are

periodic and anti periodic on the cell boundary oV, respectively. The superscripts m and p denote the matrix and the particles, respectively.

The macroscopic stress r, given by r ¼ hri, is axially symmetric because of the rotational invariance of these equations along the axis e

3

. It reads r ¼ r

m

i þ

²₃

ir

eq

^ e where r

m

and r

eq

are the overall hydrostatic and von Mises equivalent stresses, given by

r

m

¼ r

33

þ 2r

11

3 and r

eq

¼ jr

33

r

11

j ; ð5Þ

and where i is equal to +1 or 1. In general, the overall response of the nonlinear composite is thus charac- terized by the two-dimensional relation

ðr

m

; r

eq

Þ ¼ ~ f ðe

eq

; e

m

Þ: ð6Þ

If the local constitutive relations were positively homogeneous functions of the same degree m, the local strain, solution of the system of equations (4), would be a positively homogeneous function of degree one of the im- posed strain e, the local stress a positively homogeneous function of degree m and so would be the overall stress (Ponte Castan˜eda and Suquet, 1998). The general relation (6) could then be cast either in the form

r

_m

¼ ~ ~ r

0

ðs

e

Þ e

eq

e

0 m

r

eq

¼ ~ r

0

ðs

e

Þ e

eq

e

0 m

ð7Þ

or

r

m

¼ ^ r

0

ðs

e

Þ e

m

e

0 m

r

eq

¼ ^ ^ r

0

ðs

e

Þ e

m

e

0 m

: ð8Þ

Since the constitutive phases are compressible and their elastic shear moduli are finite, the local constitutive relations never exhibit exactly such homogeneity relations, even when the threshold stresses r

^r_y

vanish. How- ever, such homogeneity relation are asymptotically true for large strains, when the linear parts of the defor- mation (

_3k^rmr

and

^r_3l^eqr

e

) tend to be negligible with respect to the nonlinear one (e

0

ð

^r_r^eqr 0

Þ

ⁿ

).

For a reinforced composite or a two-phase material, the macroscopic behavior may be considered as almost incompressible, since the bulk moduli k

^r

will be chosen sufficiently large. As a consequence, the sole overall quantity of interest is the deviatoric response r

eq

¼ f ~ ðe

eq

Þ, which is characterized by the single scalar ~ r

0

when the threshold stresses vanish. On the other hand, for the porous case, the effective response is compressible, even if the matrix is almost incompressible, and is characterized by the two-dimensional relation (6), which may asymptotically restrict to the pair of functions ^ r

0

ðs

e

Þ; ~ r

0

ðs

e

Þ as illustrated in Section 4. The comparison between the exact nonlinear solution and the estimates provided by various nonlinear homogenization schemes can be performed on these macroscopic quantities, seen as functions of the parameters of the nonlin- ear problem. In Section 4, the dependance of the overall response with respect to the volume fraction f

^p

of the inclusions and the work-hardening parameter m will in particular be thoroughly investigated.

In addition to such macroscopic quantities, the comparisons can also be performed on predictions of local

fields. Of course, one should not expect a same level of accuracy of the evaluated homogenization theories in

terms of local field predictions as in terms of effective properties. However, as such models might be used for

micromechanical applications, in which the local field predictions might serve to predict the activation of local

physical mechanisms, such as damage or other microstructure evolutions, it is useful to assess the predictive

capabilities of these models even at the local scale, mostly in terms of statistics. More precisely, one can inves-

tigate the so-called inter-phase heterogeneity of the stress and strain fields, characterized by the first-order

moments e

^r

¼ hei

^r

and r

^r

¼ hri

^r

, as well as the intra-phase heterogeneities, given by the second-order moments

he ei

^r

and hr ri

^r

, or the variances C

^r_e

¼ hðe e

^r

Þ ðe e

^r

Þi

^r

¼ he ei

^r

hei

^r

hei

^r

and

C

^r_r

¼ hðr r

^r

Þ ðr r

^r

Þi

^r

¼ hr ri

^r

hri

^r

hri

^r

, which quantify the fluctuations of the local fields in the

phases with respect to their average. Because of the rotational invariance of the nonlinear homogenization

problem, the first-order moments can be decomposed into their spherical and deviatoric parts, the latter being

proportional to eˆ; they are thus characterized by two scalars. Similarly, the second-order moments and vari-

ances are transversely isotropic fourth-order symmetric tensors, which are in general characterized by 5 scalars

(see for instance Walpole (1981)). Among these, the following quantities are of special interest since, as will be recalled later, several linearization schemes are based on them

e

^r_eq

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

3 K < he ei

^r

r

¼ ffiffiffiffiffiffiffiffiffiffiffi he

²_eq

i

^r

q

; e

^r_k

e

^r_eq

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 E < C

^r_e

r

; e

^r_?

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 F < C

^r_e

r

; ð9Þ

where E ¼

²₃

^ e ^ e and F = K E are the classical fourth-order projectors (Ponte Castan˜eda, 1996) relative to the directions parallel and perpendicular to the overall load. A thorough evaluation of the linearization schemes at the local scale is out of the scope of the present paper which is focussed on the comparisons at the macroscopic scale. A few results relative to these quantities will however be presented at the end of Section 4. More detailed comparisons could also be performed in terms of statistical distribution functions (as in Bornert et al., 1994) or even in terms of the spatial distribution of local fields in the form of strain maps on the unit cell. The reader is invited to refer to Rekik (2006) for such additional results. It is noted that such comparisons are possible within the proposed methodology, since the local fields can be computed exactly both in the nonlinear composite and in the LCC; in more traditional approaches in which the LCC is homog- enized by means of a mean-field linear model, only the overall properties and the per phase first and second- order moments of the local fields are available. At this stage it is useful to recall that the local predictions of the homogenization schemes are here assumed to be given by the local fields in the LCC. This is consistent with most evaluated formulations; exceptions will be indicated later.

3. Implementation

We address here more technical issues relative to the computation of the reference exact nonlinear solution (Section 3.1) and the homogenization of the linear elastic (Section 3.2.1) or thermoelastic (Section 3.2.2) LCC.

The main assumptions of the evaluated linearization schemes are recalled in Section 3.3. and Section 3.4 is a short discussion on the accuracy of the numerical results. Let us before recall that, because of the rotational invariance along the third axis of all the data of the nonlinear problem (cell geometry, isotropic nonlinear local constitutive laws, macroscopic imposed strain), all quantities derived from these data will exhibit the same invariance. More specifically,

• Second-order tensors, such as the overall stress and per phase averages of stress and strain, admit the decomposition

c ¼ c

_m

i þ c

_d

^ e; ^ e ¼ e

₃

e

₃

¹₂

ðe

1

e

₁

þ e

₂

e

₂

Þ; ð10Þ where c

m

and c

d

are (positive or negative) scalars. If c is a strain, jc

d

j = c

eq

while j c

_d

j ¼

²₃

c

_eq

if c is a stress.

• Symmetric second-order tensor fields (such as local stress and strain fields) are axisymmetric

c ð x Þ ¼ c

_rr

ð r; z Þ e

_r

e

_r

þ c

_rz

ð r; z Þð e

_r

e

_z

þ e

_z

e

_r

Þ þ c

_zz

ð r; z Þ e

_z

e

_z

þ c

_hh

ð r; z Þ e

_h

e

_h

ð11Þ and so are first-order tensor fields like displacements

uðxÞ ¼ u

r

ðr; zÞe

r

þ u

z

ðr; zÞe

z

: ð12Þ

• Fourth-order tensors, such as the aforementioned covariance tensors, are transversely isotropic and admit the decomposition

C ¼ aE

L

þ bJ

T

þ cF

⁰

þ c

^0T

F

⁰

þ dK

T

þ d

⁰

K

_L

: ð13Þ The adopted notation is that of Bornert and Suquet (2001a), which is derived from the decomposition of transversely isotropic tensors introduced by Walpole (1981). The definition of the tensors E

L

, J

T

, F

⁰

, K

T

and K

L

, is recalled in Appendix A.

Such decompositions, which are quite obvious for the nonlinear problem, hold true for the quantities asso-

ciated with the LCC as well. Indeed the linearization procedures of all the considered schemes do not intro-

duce any additional preferential direction so that the LCC retains the invariance properties of the nonlinear

problem. In particular, the tensors of moduli of the phases admit decomposition (13), with c = c

⁰

. In addition,

as shown later, the determination of the relevant effective properties of the LCC can be performed with the application of overall axisymmetric strain tensors similar to Eq. (3).

3.1. Reference solution

The reference nonlinear solution is obtained by solving system (4) with standard finite element techniques.

Because of the rotational invariance, the displacement field admits decomposition (12) and the stress and strain fields can be written as in Eq. (11), so that two-dimensional axisymmetric elements can be used. Because of the additional symmetry with respect to the transverse plane, only the domain (r, z) 2 V

2

= [0, R] · [0, H]

has to be meshed, the origin of the coordinates being centered on the inclusion. The symmetry and periodicity conditions reduce then to the simple set of equations (see Tvergaard (1982); Koplik and Needleman (1988) for details)

u

_r

ðR; zÞ ¼ Re

11

and u

_r

ð0; zÞ ¼ 0; 0 6 z 6 H ; u

_z

ð r; H Þ ¼ He

33

and u

_z

ð r; 0Þ ¼ 0; 0 6 r 6 R;

ð14Þ

with e

11

¼ e

_m

e

eq

=2 and e

33

¼ e

_m

þ e

eq

: ð15Þ

The FE code CAST3M has been used for the determination of the local fields. An appropriate iterative pro- cedure is used to address the nonlinearity of the system.

After convergence at an imposed macroscopic strain, the global stress components are obtained by averaging

r

11

¼ r

22

¼

_jV^4p_j

R

R 0

R

H

0

ðr

rr

ðr; zÞ þ r

hh

ðr; zÞÞr dr dz ¼

^hrrrþr₂ ^hhi

; r

33

¼

_jV^4p_j

R

R

0

R

H

0

r

zz

ðr; zÞr dr dz ¼ hr

zz

i;

8 <

: ð16Þ

the other components being zero, as already noted. The overall mean and deviatoric stresses are then obtained from Eq. (5). Similar relations hold true for the computation of the per phase average strains and stresses, with an integration restricted to the corresponding phase and jVj replaced by its volume. The scalar second-order moment e

^r_eq

is obtained by averaging the square of the local equivalent strain

ðe

^r_eq

Þ

²

¼ he

²_eq

i

^r

¼ h 2

3 ð2e

²_rz

þ ðe

rr

e

m

Þ

²

þ ðe

hh

e

m

Þ

²

þ ðe

zz

e

m

Þ

²

Þi

^r

; ð17Þ with e

m

¼

¹₃

ðe

rr

þ e

hh

þ e

zz

Þ. The parallel and perpendicular components of the (tensorial) second-order mo- ments are obtained from

E < he ei

^r

¼

²₃

hðe : ^ eÞ

²

i

^r

¼

²₃

hðe

zz

^errþe₂^hh

Þ

²

i

^r

; ð18Þ F < he ei

^r

¼

³₂

ðe

^r_eq

Þ

²

E < he ei

^r

: ð19Þ Since F :: (hei

^r

hei

^r

) = 0 and E < ðhei

^r

hei

^r

Þ ¼

²₃

ðhei

^r

: ^ eÞ

²

¼

³₂

ðe

^r_eq

Þ

²

, we finally get

e

^r_k

e

^r_eq

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4

9 hðe

zz

e

rr

þ e

hh

2 Þ

²

i

^r

ðe

^r_eq

Þ

²

r

ð20Þ

e

^r_?

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe

^r_eq

Þ

²

4

9 hðe

zz

e

rr

þ e

hh

2 Þ

²

i

^r

r

: ð21Þ

To get all the desired quantities to characterize the inter- and intra-phase heterogeneity of the strain field, one has thus to integrate over each phase domain the four scalars e

zz

, (e

rr

+ e

hh

), e

²_eq

and ðe : ^ e Þ

²

. This can be per- formed with standard operators of the FE code CAST3M. Similar relations hold for the stress field.

3.2. Linear homogenization

The equations governing the global and local response of the LCC are similar to those relative to the non-

linear problem (system 4), the only difference being that the local constitutive relation is now linear, taking the

form

rðxÞ ¼ X

r

v

^r

ðxÞðL

^r

: eðxÞ þ s

^r

Þ; 8x 2 V : ð22Þ

The fourth-order tensor L

^r

is the homogeneous tensor of moduli and the second-order tensor s

^r

is the uniform polarization of phase r. Their values depend on the linearization scheme under consideration, as detailed in the next section. However, because of the global symmetry of the problem, these quantities are invariant with re- spect to any rotation along the symmetry axis. The tensor L

^r

then reads

L

^r

¼ a

L^r

E

_L

þ b

_L^r

J

_T

þ c

_L^r

ðF

⁰

þ

^T

F

⁰

Þ þ d

L^r

K

_T

þ d

⁰_Lr

K

_L

: ð23Þ The following results are given under the assumption that L

^r

takes this general form. Indeed, as recalled later, the actual tensors defined by the evaluated linearization schemes admit the decomposition (Ponte Castan˜eda, 1996)

L

^r

¼ 3k

^r

J þ 2k

^r

E

^r

þ 2l

^r

F

^r

; ð24Þ

where E

^r

¼

²₃

^ e

^r

^ e

^r

, F

^r

= K E

^r

and ^ e

^r

¼ ð K : e

^r

Þ =e

eq

. Since e

^r

is axially symmetric, we have K : e

^r_d

¼ e

^r_eq

^ e and therefore

^ e

^r

¼ K : e

^r

e

^r_eq

¼ K : e

e

eq

¼ ^ e; E

^r

¼ E ¼ 2

3 ^ e ^ e; F

^r

¼ F ¼ K E: ð25Þ

The tensor L

^r

is therefore transversely isotropic and can be written in the more general form (23) with the fol- lowing relations between the constants (k

^r

, k

^r

, l

^r

) and ða

L^r

; b

_L^r

; c

_L^r

; d

L^r

; d0

L^r

Þ

a

_L^r

¼ k

^r

þ

⁴₃

k

^r

; b

_L^r

¼ 2k

^r

þ

²₃

k

^r

; c

_L^r

¼ c

⁰_Lr

¼ ffiffiffi

p 2

ðk

^r

²₃

k

^r

Þ;

d

_L^r

¼ d

⁰_Lr

¼ 2l

^r

: 8 >

> >

> <

> >

> >

:

ð26Þ

For some linearization schemes, L

^r

is isotropic. For these, the previous relations still hold, with the simplifi- cation k

^r

= l

^r

.

The polarization s

^r

vanishes for all linearization schemes that define a linear elastic LCC. For all other schemes, s

^r

is axially symmetric as a consequence of the general symmetry properties of the considered prob- lem. In addition, it turns out that the polarization s

^r

is trace free, because the considered local nonlinear con- stitutive equations are linear on their spherical part. The tensor s

^r

then takes the form

s

^r

¼ 2

3 s

^r_d

^ e; ð27Þ

where js

^r_d

j is the von Mises equivalent stress of s

^r

.

The tensors L

^r

and s

^r

are ‘‘inputs’’ defining the LCC. The other inputs are its microgeometry, which, con- sistently with the presented methodology, is exactly the same as the microstructure of the nonlinear composite and can be represented by exactly the same cylindrical unit cell under periodic boundary conditions. The third input is the overall load applied to the LCC. A full homogenization of the LCC would require to investigate the global and local responses of the LCC under any load. However, all required ‘‘outputs’’ turn out to be obtained with axially symmetric overall strains. As a consequence, the same 2D axisymmetric cell, with the boundary conditions (14), can be used again. More precisely, the expected ‘‘outputs’’ are the overall stress induced in the LCC by the overall strain imposed on the nonlinear composite, the first-order moment of the strain in each phase of the LCC, which, as previously, admits the decomposition (10), and the traces (9) of the second-order moments or the variances of the strain fields. Similar stress-like quantities might also be useful. Note that these local quantities are of interest, first for comparison purposes as was already the case for the nonlinear homogenization problem, but also in order to define the constitutive relations of the LCC (i.e., L

^r

and s

^r

), according to the principles of the linearization scheme under consideration, in an iterative and (hopefully) converging procedure.

In principle, since linear elastic constitutive relations are particular cases of nonlinear ones, all the relations

given in Section 3.1 apply to the LCC. Because of the linearity, additional properties apply. They are detailed

in the following two sections, devoted first to the case s

^r

= 0 and then to the general thermoelastic situation. It is in particular shown how the simpler linear elastic results can be efficiently extended to the second case, with- out additional computations.

3.2.1. Elastic case

The linearity of the local problem ensures the existence of the localization tensor field A(x) and its per phase averages

eðxÞ ¼ AðxÞ : e; ð28Þ

e

^r

¼ hei

^r

¼ A

^r

: e where A

^r

¼ h A i

^r

: ð29Þ

The tensors A

^r

are fourth-order tensors, not necessarily symmetric, which depend on the microstructure and on the local properties, but not on the overall load. They are transversely isotropic and read

A

^r

¼ a

A^r

E

_L

þ b

_A^r

J

_T

þ c

_A^r

F

⁰

þ c

⁰_A^rT

F

⁰

þ d

A^r

K

_T

þ d

⁰_A^r

K

_L

ð30Þ so that the localization relation (29) takes the form

e

^r₃₃

e^r₁₁þe^r₂₂

ffiffi

2 p e^r₁₁e^r₂₂

ffiffi

2 p

ffiffiffi 2 p e

^r₁₂

ffiffiffi 2 p e

^r₁₃

ffiffiffi 2 p e

^r₂₃

0 B B B B B B B B B B

@

1 C C C C C C C C C C A

¼

a

A^r

c

⁰_Ar

0 0 0 0 c

_Ar

b

_Ar

0 0 0 0

0 0 d

A^r

0 0 0

0 0 0 d

A^r

0 0

0 0 0 0 d

⁰_Ar

0

0 0 0 0 0 d

⁰_Ar

0 B B B B B B B B

@

1 C C C C C C C C A

e

33 e11^pþe

ffiffi

₂22 e₁₁^pe

ffiffi

₂22

ffiffiffi 2 p e

12

ffiffiffi 2 p e

13

ffiffiffi 2 p e

23

0 B B B B B B B B B @

1 C C C C C C C C C A

: ð31Þ

Note that the coefficient of the localization tensor of one phase are deduced from those of the other by the relation

X

r¼m;p

f

^r

A

^r

¼ I; ð32Þ

where the f

^r

= hv

^r

i denote the volume fractions of the phases. The overall stress is obtained from r ¼ hri ¼ hL : Ai : e ¼ X

r¼m;p

f

^r

L

^r

A

^r

: e: ð33Þ

The tensor of effective moduli L ~ ¼ hL : Ai admits the decomposition (13) with coefficients deduced from Eqs.

(33), (31) and (23)

~ a ¼ P

r

f

^r

ða

A^r

a

_L^r

þ c

_Ar

c

⁰_Lr

Þ;

b ~ ¼ P

r

f

^r

ðc

⁰_Ar

c

_L^r

þ b

_A^r

b

_L^r

Þ;

~ c ¼ P

r

f

^r

ða

A^r

c

_L^r

þ c

_A^r

b

_L^r

Þ;

~ c

⁰

¼ P

r

f

^r

ðc

⁰_A^r

a

_L^r

þ b

_Ar

c

⁰_Lr

Þ;

~ d ¼ P

r

f

^r

ðd

A^r

d

L^r

Þ;

~ d

⁰

¼ P

r

f

^r

ðd

⁰_Ar

d

⁰_Lr

Þ:

8 >

> >

> >

> >

> >

> >

> >

> >

> <

> >

> >

> >

> >

> >

> >

> >

> >

:

ð34Þ

From Eqs. (31), (32) and (34), it is clear that the derivation of the overall stress and the per phase first-order moments of the strain induced by an axisymmetric overall deformation only requires the determination of the four scalars ða

A^r

; b

_A^r

; c

_A^r

; c

⁰_Ar

Þ relative to one of the constituents. This can easily be achieved by solving the localization problem with the FE tool for two independent axisymmetric loads. For instance e ¼ e

33

e

₃

e

₃

gives a

_A^r

¼ he

zz

i

^r

and c

_Ar

¼ he

rr

þ e

_hh

i

^r

= ffiffiffi

p 2

while e ¼ e

₁

e

₁

þ e

₂

e

₂

leads to b

_Ar

¼ he

rr

þ e

_hh

i

^r

=2 and

c

⁰_Ar

¼ he

zz

i

^r

= ffiffiffi p 2

. Alternatively, one may also choose the elementary loads e ¼ i and e ¼ ^ e. In the sequel, the corresponding local fields will be called e

^e¼i

ðxÞ ¼ AðxÞ : i and e

^{e¼^e}

ðxÞ ¼ AðxÞ : ^ e, respectively. The relations giv- ing the constants ða

A^r

; b

_Ar

; c

_Ar

; c

⁰_Ar

Þ from the averages of e

^e¼i

and e

^{e¼^e}

over phase r are easily obtained but are not given here for brevity. Note that the computation of these quantities requires the construction of one stiffness matrix, its inversion, two matrix products and four spatial integrations of scalar quantities.

To obtain the traces (9), additional computations are required as described hereafter. The local field under the general axisymmetric load ðe

m

; e

eq

Þ is given by superposition as

eðxÞ ¼ e

m

e

^e¼i

ðxÞ þ e

eq

e

^{e¼^e}

ðxÞ: ð35Þ

Its second-order moment in phase r is given by

he ei

^r

¼ e

²_m

he

^e¼i

e

^e¼i

i

^r

þ e

²_eq

he

^{e¼^e}

e

^{e¼^e}

i

^r

þ e

m

e

eq

ðhe

^{e¼^e}

e

^e¼i

i

^r

þ he

^e¼i

e

^{e¼^e}

i

^r

Þ; ð36Þ so that e

^r_eq

reads

e

^r_eq

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e

²_m

hðe

^e¼i_eq

Þ

²

i

^r

þ e

²_eq

hðe

^{e¼^}_eq^e

Þ

²

i

^r

þ

⁴₃

e

m

e

eq

he

^{e¼^e}

: K : e

^e¼i

i

^r

q

: ð37Þ

Similarly, we have 3

2 E < he ei

^r

¼ hððe

m

e

^e¼i

þ e

eq

e

^{e¼^e}

Þ : ^ eÞ

²

i

^r

¼ e

²_m

hðe

^e¼i

: ^ e Þ

²

i

^r

þ e

²_eq

hðe

^{e¼^e}

: ^ e Þ

²

i

^r

þ 2e

m

e

eq

hðe

^e¼i

: ^ e Þðe

^{e¼^e}

: ^ e Þi

^r

; ð38Þ

F < he ei

^r

¼ K < he ei

^r

E < he ei

^r

ð39Þ

from which the quantities e

^r_k

and e

^r_?

can be deduced as was done at the end of Section 3.1. This shows that six additional integrations are then required, among which two involve quantities that couple the local fields in- duced by two different loads. Again this can be performed by the object-oriented operators of CAST3M.

3.2.2. Thermoelastic extension

The previous relations can be extended to situations with s

r

5 0. The local strain field and their averages read now

eðxÞ ¼ AðxÞ : e þ aðxÞ; ð40Þ

e

^r

¼ A

^r

: e þ a

^r

with A

^r

¼ hAi

^r

and a

^r

¼ hai

^r

; ð41Þ where A(x) coincides with the localization tensor field of the elastic LCC introduced in Section 3.2.1. The sec- ond-order tensor field a(x) is the local strain under vanishing overall strain. Since we are dealing with two- phase composites, the so-called Levin’s relations apply (Levin, 1967) and allow to compute a(x) from A(x)

aðxÞ ¼ ðAðxÞ IÞ : ðDLÞ

¹

Ds where DL ¼ ðL

²

L

¹

Þ and Ds ¼ s

²

s

¹

ð42Þ so that

eðxÞ ¼ AðxÞ : C B where B ¼ ðDLÞ

¹

Ds and C ¼ e þ B ¼ e þ ðDLÞ

¹

Ds: ð43Þ In addition, the overall stress-strain relation reads

r ¼ ~ L : e þ ~ s where L ~ ¼ X

r

f

^r

L

^r

: A

^r

and ~ s ¼ X

r

f

^r

s

^r

: A

^r

: ð44Þ

We refer to Willis (1983) for details on such derivations.

The computation of ~ L : e can be performed exactly as in the purely elastic case and requires only the four constants ða

A^r

; b

_A^r

; c

_A^r

; c

⁰_Ar

Þ relative to one of the constituents. Since s

^r

is proportional to eˆ, the computation of s

^r

: A

^r

requires the same four constants. The overall stress induced by a given macroscopic strain is thus obtained from the same quantities as those required in the elastic case.

For the per phase first-order moment of the strain field, we have

e

^r

¼ A

^r

: C B: ð45Þ

Since Ds is proportional to eˆ and L

^r

takes the form L

^r

= 3k

^r

J + 2k

^r

E + 2l

^r

F, the product (DL)

¹

Ds will also be proportional to ^ e, so that B and C read B = B

eq

eˆ (B

eq

being positive or negative) and C = C

m

i + C

eq

eˆ with C

m

¼ e

m

and C

eq

¼ e

eq

þ B

eq

. The per phase first-order moments of the strain are then obtained from Eq.

(45), which requires again the four constants ða

A^r

; b

_Ar

; c

_Ar

; c

⁰_Ar

Þ obtained as in the elastic case.

Traces of the second-order moments are obtained with relations similar to Eqs. (37) and (39) and require the same additional integrations. More specifically

he ei

^r

¼ C

²_m

he

^e¼i

e

^e¼i

i

^r

þ C

²_eq

he

^{e¼^e}

e

^{e¼^e}

i

^r

þ B

²_eq

^ e ^ e þ C

_m

C

eq

ðhe

^{e¼^e}

e

^e¼i

i

^r

þ he

^e¼i

e

^{e¼^e}

i

^r

Þ

C

_m

B

_eq

ðhe

^e¼i

i

^r

^ e þ ^ e he

^e¼i

i

^r

Þ C

_eq

B

_eq

ðhe

^{e¼^e}

i

^r

^ e þ ^ e he

^{e¼^e}

i

^r

Þ: ð46Þ The computation of the traces along K, E and F requires the same six quadratic integrals already required in the elastic case: hðe

^e¼i_eq

Þ

²

i

^r

, hðe

^{e¼^}_eq^e

Þ

²

i

^r

, hðe

^e¼i

: ^ eÞ

²

i

^r

, hðe

^{e¼^e}

: ^ eÞ

²

i

^r

, he

^e¼i

: K : e

^{e¼^e}

i

^r

and he

^e¼i

: E : e

^{e¼^e}

i

^r

, as well as the two averages he

^e¼i

: ^ ei

^r

and he

^{e¼^e}

: ^ ei

^r

, which derive from the four constants ða

A^r

; b

_A^r

; c

_A^r

; c

⁰_Ar

Þ. This shows that, even if the final expressions are slightly more complex, the numerical complexity of a linearization scheme based on a thermoelastic LCC is identical to that of a scheme using a linear LCC.

3.3. Nonlinear formulations

In this section, we describe the different nonlinear formulations which provide the elastic and thermoelastic LCC and spell out how to solve the whole nonlinear system consisting of both the linear (thermo)elastic homogenization problem and the linearization procedure.

The various linearization procedures can be split into two categories. The first consists of the nonlinear extensions which define a LCC where the phase behavior is described by a local (thermo)elastic stress-strain relation r(x) = L

^r

(e

^r

): e(x) (+s

^r

(e

^r

)). For some linearization procedures such as the secant or affine approaches, the reference strain e

^r

is set to the average of the strain field over the phase r, namely e

^r

¼ e

^r_eq

. For other for- mulations such as the secant modified extension—i.e., the variational procedure —, the strain field fluctuations are taken into account in such a way that the proposed prescription for the reference strain becomes e

^r

¼ e

^r_eq

. For those linearization procedures which we will refer to as the ‘‘stress-strain approaches’’, the macroscopic stress r is defined as the mean value of the local stress field, say r ¼ hri, computed according to Eq. (44).

For the second class of linearization procedures which we will refer to as the ‘‘potential based approaches’’, the effective stress is defined according to Hill’s theorem as the derivative of the effective potential with respect to the effective strain r ¼

^o~_oe^w

ðeÞ. The mathematical form of the overall potential depends on the chosen line- arization procedures. Generally, the main idea is to define a thermoelastic LCC whose effective potential can be used to estimate the effective potential of the nonlinear composite. The thermoelastic strain potential of the thermoelastic LCC reads

w

_T

ðx; eÞ ¼ X

r

v

^r

ðxÞw

^r_T

ðeÞ; ð47Þ

where the thermoelastic strain potentials w

^r_T

ðeÞ are second-order Taylor-type expansions (or third-order expansion for the LS approach) defined as

w

^r_T

ðeÞ ¼ w

^r

ðe

^r

Þ þ ow

^r

oe ðe

^r

Þ : ðe e

^r

Þ þ 1

2 ðe e

^r

Þ : L

^r

: ðe e

^r

Þ: ð48Þ

In Eq. (48), w

^r

(e) is the nonlinear strain-energy function of the phase r of the nonlinear composite. The ref-

erence strains e

^r

and the moduli L

^r

are constant inside each phase r. Those tensors are chosen such that they

generate the best possible estimates of the nonlinear effective potential w ~ from known estimates of the ther-

moelastic effective potential w ~

_T

. In general, the optimal values of e

^r

and L

^r

are derived from stationarity con-

ditions. For the initial second-order method and the LS procedure, L

^r

¼ ðo

²

w=oe

²

Þðe

^r

Þ. For the second-order

modified procedure, the expression of the tensors L

^r

is more complicated (Eq. (52)). As mentioned above, the

macroscopic constitutive behavior of the potential based approaches is evaluated by the derivative of the effec-

tive potential with respect to the effective strain. Since both the effective stress and strain are axially symmetric,

the effective behavior can be shown to be defined, as in the purely isotropic case, by two scalar equations

r

m

¼ o w ~ oe

m

ðeÞ and r

eq

¼ o w ~ oe

eq

ðeÞ; ð49Þ

where the derivatives are found numerically.

3.3.1. The nonlinear problem

Before presenting in more details the various nonlinear formulations, we first focus our attention on the whole nonlinear system to be solved. For all the linearization procedures but the modified second-order approach (Ponte Castan˜eda, 2002), the nonlinear problem can be written in the following form

uðxÞ ¼ e:x þ u

ðxÞ 8x 2 V and u

# on oV

eðuðxÞÞ ¼

¹₂

ðruðxÞ þ

^t

ruðxÞÞ ¼ e þ eðu

ðxÞÞ; 8x 2 V divðrÞ ¼ 0; 8 x 2 V and r n # on oV

rðxÞ ¼ P

r¼m;p

v

^r

ðxÞðL

^r

: eðxÞ þ s

^r

Þ

9 >

> >

> >

=

> >

> >

> ;

local linear problem

L

^r

¼ L

^r

ðe

^r

Þ ; s

^r

¼ s

^r

ðe

^r

Þ e

^r

¼ hei

^r_eq

or ffiffiffiffiffiffiffiffiffiffiffi

he

²_eq

i

^r

q

)

nonlinear relations 8 >

> >

> >

> >

> >

> >

<

> >

> >

> >

> >

> >

> :

ð50Þ

where L

^r

(e) and s

^r

(e) are known functions whose exact expressions depend on the chosen linearization proce- dure. To solve this system of equations, a fixed-point iterative procedure is used. An initial value denoted p

^r₀

is ascribed to the reference strain e

^r

for each phase. From Eq. (50e), we obtain the starting values of the moduli tensors L

^r

and the polarization s

^r

. Then, the numerical procedures described in Sections 3.2.1 and 3.2.2 are applied to provide a first evaluation of the first e

^r_eq

and second-order e

^r_eq

moments of the strain field in the LCC. According to the chosen linearization procedure, Eq. (50f) provides a new evaluation of the reference strain denoted p

^r₁

. This iterative procedure goes on until the discrepancy p

^r₁

p

^r₀

meets the convergence crite- rion. Generally, three or four iterative steps are sufficient to reach convergence. For the modified second-order procedure, the nonlinear problem is solved by the same numerical scheme except that the moduli L

^r

, as shown in Eq. (52), depend on two reference strains e

^r

and ^ e

^r

.

3.3.2. Stress–strain approaches

Classical and modified secant formulations. For both formulations, the LCC is elastic (s

^r

= 0). The tensor L

^r

is defined as the isotropic tensor of secant moduli evaluated at the reference strain e

^r

, namely L

^r

¼ 3k

^r

J þ 2l

^r_sct

ðe

^r

ÞK where l

^r_sct

ðe

eq

Þ ¼ r

eq

ðe

eq

Þ=3e

eq

is the secant shear modulus. For the Ramberg–Osgood constitutive law, e

eq

is related to r

eq

through Eq. (2b) and r

eq

= f

r

(e

eq

) is derived by means of a numerical inversion procedure. For the classical secant formulation SEC, the reference strain is prescribed to be the per phase average of the strain field in the LCC—that is e

^r

¼ e

^r_eq

—whereas for the modified secant extension VAR, which coincides with the variational approach of Ponte Castan˜eda (1991) (Suquet, 1995), the reference strain becomes e

^r

¼ e

^r_eq

. It is worth recalling that the variational procedure provides an upper bound for the effective energy.

Affine formulations. For the original affine approach referred to as AFF-ANI (Masson et al., 2000), the lin- ear constitutive behavior of the individual constituents of the LCC follows a thermoelastic constitutive law where the elastic moduli and the polarization tensors are defined by L

^r

¼ L

^r_tgt

ðe

^r

Þ ¼

^o_oe^2w2^r

ðe

^r

Þ and s

^r

¼

^ow_oe^r

ðe

^r

Þ L

^r

: e

^r

, respectively. Note that the thermoelastic law of each individual constituent of the LCC is tangent to the constitutive law of each phase of the real nonlinear composite r ¼

^ow_oe^r

ðeÞ at e ¼ e

^r

. Unlike the secant methods, the tensor L

^r

is transversely isotropic and can be written as L

^r

¼ 3k

^r

J þ 2l

^r_sct

ðe

^r_eq

ÞF þ 2l

^r_tgt

ðe

^r_eq

ÞE where l

^r_tgt

ðe

eq

Þ ¼

_3de^dreq

eq

ðe

eq

Þ is the tangent shear modulus. Due to the anisot-

ropy of L

^r

, applying the affine formulation is less straightforward than the isotropic secant procedures. To get

round this drawback, two simplified isotropic versions have been proposed. In the first variant referred to as

AFF-ISOT (Chaboche and Kanoute´, 2003), L

^r

is defined by L

^r

¼ 3k

^r

J þ 2l

^r_tgt

ðe

^r_eq

ÞK. In the second variant

AFF-ISOI, it is defined as the projection of the actual transversely isotropic tangent tensor on the subspace

of the isotropic tensors, namely L

^r

¼ 3k

^r

J þ 2l

^r_inv

ðe

^r_eq

ÞK with l

^r_inv

ðe

^r_eq

Þ ¼

^{4lrsctðereqÞþl}₅ ^rtgtðereqÞ

.